JP2024507834A - L0 regularization-based compression detection system and method using coherent Ising machine - Google Patents

Abstract

A system and method for L0 regularization-based compressed sensing (CS) may use a quantum-classical hybrid system consisting of a coherent Ising machine (CIM) and a classical digital processor (CDP). CIM and CDP each implement alternating minimization for L0 regularization-based compression sensing (CS). A truncated Wigner stochastic differential equation (W-SDE) is obtained from the master equation for the density operator of the network of degenerate optical parametric oscillators.

Description

CROSS-REFERENCE This application claims priority to U.S. Provisional Application No. 63/151,441, filed February 19, 2021, the entirety of which is incorporated herein by reference.

Appendix Appendix A (page 7) contains supplementary material and figures that form part of this specification and are incorporated herein by reference.

Appendix B (page 10) lists the methods used for various calculations herein, which appendix forms part of this specification and is incorporated herein by reference.

The present disclosure relates to systems and methods for compressing sensing using coherent Ising machines.

Various techniques currently exist for performing simulation and optimization processes using quantum devices or machines. These quantum devices or machines are very complex and difficult/expensive to build. Examples of quantum machines include Google's quantum computers and IBM's quantum computers and D-WAVE's quantum annealer. All of these quantum machines are extremely expensive and difficult to build and operate.

The least absolute shrinkage and selection operator (LASSO) is extremely useful for solving a variety of sparse signal reconstruction problems in exploration geophysics, magnetic resonance imaging, black hole observation, and materials informatics. This is an efficient approach. The LASSO operator is
(1)
where x is the N-dimensional source signal, y is the M-dimensional observation signal, A is the M×N observation matrix, and λ is the regularization parameter.

L1 regularization-based compressed sensing (CS), including LASSO, can be formulated as a convex optimization problem, for which many efficient heuristic algorithms are available. Equation (1) above is asymptotically equal to the L1 minimization base CS (minimize ||x|| ₁ such that y=Ax) in the limit λ→+0, and therefore this limit , LASSO is such that the ratio of the number of non-zero elements of x to N, i.e., the sparseness a, is less than a critical value a _c (a _c is the measured compression ratio α (defined as α = M/N)) A sparse source signal can be reconstructed with infinitesimal error as long as it is smaller.

On the other hand, the L0 regularization-based CS can be formulated with the following L0 norm instead of the L1 norm:
(2)

It has been suggested that L0 regularization-based CS may outperform L1 regularization-based CS. This is because L1 regularization imposes a contraction on variables that exceed a certain threshold (soft-thresholding), whereas L0 regularization does not impose such a contraction (hard thresholding). It is. Furthermore, equation (2) is asymptotically equal to the L0-minimizing base CS (minimizing ||x|| ₀ such that y=Ax) in the limit λ→+0, so in this limit, The L0 regularization-based CS can be expected to reconstruct x with infinitesimal error as long as a is smaller than the critical value a _c =α. The number of non-zero elements in x is the effective rank of the system of linear equations, so any system will exceed its performance limit when a>α, since it will not have a single unique solution. Note that this is not possible.

The L0 regularization-based CS defined by equation (2) can be equivalently reformulated as a two-fold optimization problem:
(3)

Here, the element r _i in r indicates the real value of the i-th element in the N-dimensional source signal. Vector σ is called a support vector indicating the location of non-zero elements of the N-dimensional source signal. The element σ _i in σ takes 0 or 1 to indicate whether the i-th element of the source signal is zero or non-zero. The symbol ° indicates the Hadamard product. From the element-wise expression of equation (3), the Hamiltonian (H or cost function) of the L0 regularization base CS is:
(Four)
, where,
is an element in the M×N observation matrix A, and y ^u is an element in the M-dimensional observation signal.

Under the assumption that the number of ones in the support vector σ is less than or equal to M, the set of simultaneous linear equations obtained from equation (3) in terms of r is give the solution. On the other hand, minimizing H with respect to σ is identical to minimizing the Ising Hamiltonian given r. Interaction
induces frustration among the spins σ _i , so the Hamiltonian may have many local minima. Therefore, the L0 regularization base CS cannot be formulated as a convex optimization.

It would be desirable to provide a system and method for performing L0 regularization-based compression sensing without the difficulty of estimating support vectors, and it is to this purpose that this disclosure is directed.

The systems and methods herein may be used for L0 regularization-based compression sensing using a quantum-classical hybrid system comprised of a quantum machine and a classical digital processor (CDP).

In one embodiment, a hybrid system for L0 regularization-based compression sensing of a source signal is provided. The system includes a quantum machine configured to optimize a first parameter associated with the source signal to minimize a cost function and a second parameter associated with the source signal to minimize the cost function. and a classical machine configured to optimize the parameters of.

In another embodiment, a method for L0 regularization-based compression sensing of a source signal may be provided. The method includes optimizing a first parameter associated with the source signal by a quantum machine to minimize a cost function, and optimizing a second parameter associated with the source signal to minimize the cost function. optimizing by a classical machine.

In yet another embodiment, a method of implementing L0 regularization-based compression sensing may be provided. The method may include injecting a plurality of pump pulses into a coherent Ising machine optical parametric oscillator having an optical parametric oscillator formed within a fiber ring cavity having an output coupler and an input coupler, the output coupler being a homodyne It communicates with the detection output and second harmonic generation (SHG) crystal.

FIG. 1 illustrates a general quantum-classical hybrid system composed of a quantum machine and a classical digital processor (CDP) capable of performing L0 regularization-based compression sensing. 1 is a diagram illustrating a quantum-classical hybrid system composed of a coherent Ising machine (CIM) and a CDP capable of performing L0 regularization-based compression sensing; FIG. FIG. 4 compares solutions of methods for L0 regularization-based compression sensing; Figure 3 is a phase diagram of L0 regularization based compression sensing and LASSO for various cases. FIG. 4 illustrates a basin of attraction for L0 regularization-based compression sensing depending on an initial threshold. FIG. 3 shows root mean square error (RMSE) values for L0 regularization-based compression sensing and LASSO for a half-Gaussian source signal. FIG. 3 shows root mean square error (RMSE) values for L0 regularization-based compression sensing and LASSO for a Gaussian source signal. FIG. 6 illustrates the performance of L0 regularization-based compression sensing on example data. FIG. 4 illustrates four types of probability density functions used to generate source signals in numerical experiments. 2 is a flowchart and more detailed pseudocode of a method for L0 regularization-based compression sensing. 2 is a flowchart and more detailed pseudocode of a method for L0 regularization-based compression sensing. 2 is a flowchart and more detailed pseudocode of a method for L0 regularization-based compression sensing.

The present disclosure is particularly applicable to systems and methods for L0 regularization-based compression sensing using a quantum-classical hybrid system comprised of a quantum machine and a classical digital processor (CDP). A coherent Ising machine (CIM) is a suitable quantum machine for this system. This is because this optimization problem can only be solved by a densely connected network. It is in this context that the present disclosure will be described. However, it will be appreciated that the systems and methods may be implemented in other ways. Furthermore, although the exemplary data is medical data, the systems and methods for L0 regularization-based compression sensing may be used for any type of data and are not limited to any particular type of data.

In the system and method for L0 regularization-based compression sensing, the technical problem of difficulty in estimating support vectors described above is addressed by a technical solution that is a hybrid system with a quantum machine and a CDP. FIG. 1A shows a quantum-classical hybrid system composed of a coherent Ising machine (CIM) and a CDP that can perform L0 regularization-based compression sensing. This system consists of two minimization processes; (i) a quantum machine optimizes σ to minimize H under the condition that r is fixed, and (ii) a CDP optimizes σ to minimize H under the condition that r is fixed. The two-step optimization problem is solved by alternately optimizing r in order to minimize H under the conditions.

Some quantum machines such as quantum annealer, quantum approximation optimization algorithm, CIM, etc. could possibly be used to optimize σ. A comparison of these candidates reveals that the measurement-feedback (MFB) CIM is one of the most suitable machines for this purpose. In fact, MFB-CIM can build any densely connected network consisting of degenerate optical parametric oscillators (OPOs). This is because MFB-CIM uses a time division multiplexing scheme and MFB. In contrast, QA and almost all other machines can only support local graphs, including chimera graphs, and therefore a tightly connected network for optimizing σ is similar to the Lechner-Hauke- It has to be embedded in a fixed hardware local graph by using the Zoller scheme, which requires additional physical spins. Furthermore, we show that MFB CIM experimentally outperforms quantum annealer on two problem sets (one problem is a fully connected Sherrington-Kirkpatrick model and the other problem is a dense graph MAX-CUT). It has been reported. In contrast to the exponential computation time proportional to exp(O(N ² )) for quantum annealers, CIM
), where N is the problem size.

FIG. 1B shows that the L0 regularization-based compression sensing system and method may be implemented, in one embodiment, using a quantum-classical hybrid system 100 comprised of a CIM 102 and a classical digital processor (CDP) 104. . This system 100 includes two minimization processes; (i) CIM 102 optimizes σ to minimize the Hamiltonian cost function H using a given real value r of the source signal; and (ii) CDP 104 alternately optimizes r to minimize H using a given support vector σ. Using the system, the Wigner stochastic differential equation (W-SDE) from the master equation for the density operator of the network consists of an OPO. The systems and methods implement statistical mechanical methods based on self-consistent signal-to-noise analysis (SCSNA) from which macroscopic equations for the entire system may be derived. . As discussed in detail below, the performance of the disclosed systems and methods approaches the theoretical limits of L0 minimization-based compressive sensing (CS) and possibly exceeds the theoretical limits of known LASSO techniques.

In more detail, shown in FIG. 1B, CIM 102 of system 100 has a pump pulse injected through a second harmonic generation (SHG) crystal into an optical parametric oscillator (OPO) formed within a fiber ring cavity. A periodically poled lithium niobate (PPLN) waveguide device induces a phase-sensitive degenerate optical parametric amplification of the signal pulses such that each OPO pulse moves beyond the oscillation threshold to the zero phase state (up (corresponding to spin) or π phase state (corresponding to down spin). A portion of each pulse is extracted from the main cavity by an output coupler and measured by an optical homodyne detector. A field programmable gate array (FPGA) calculates a feedback signal that is then provided to an intensity modulator (IM) and a phase modulator (PM) to For each OPO pulse through the input coupler, generate an injection field as described in equation (5) below. H(X _i ) is the binary value of the in-phase amplitude of the i-th OPO pulse, 0 or 1, which is the support estimate transferred to the CDP 104 . CDP 104 solves the linear simultaneous equation (Equation (7) discussed below) and the solution r _i is transferred to CIM 102 as shown in FIG. 1B.

Role of CIM and CDP

The CIM-CDP hybrid system 100 of FIG. 1B performs the L0 regularization-based CS described in equations (3) and (4) above. The system achieves optimization by performing the following two minimization processes alternately. CIM 102 optimizes σ to minimize H with a given r and then forwards σ to CDP 104. CDP 104 optimizes r to minimize H using a given σ and then forwards r to CIM 102. Figures 9A, 9B, and 10, discussed in more detail below, show details of the method using two alternating minimization processes. In the method 900 shown in FIGS. 9A, 9B, and 10, to widen the sphere of attraction, a heuristically linear threshold reduction is performed in which the threshold η decreases linearly from η _init to η _end as the alternating minimization progresses. be introduced.

CIM 102 estimates the support vector σ, ie, the location of non-zero elements of the source signal. To optimize σ to minimize H (Hamiltonian/cost function) with a given r, CIM 102 uses measurement feedback circuitry to determine the intensity modulator (IM) and phase modulator (PM ), both modulators generate a light injection field for the target (i-th) OPO pulse:
(Five)
Here, K is the gain of the feedback circuit and η is the threshold value. η is
In relation to the regularization parameters in equations (3) and (4) according to h _i are local fields explained below. The method uses two different functions F ₊ (h) and F _± (h) for the local field according to the source signal. F ₊ (h) is the identity function: F ₊ (h) is used for non-negative source signals. F _± (h) is an absolute value function: F _± (h) is used for signed source signals.

The local field for estimating the support vector on CIM is
(6)
where X _j is the in-phase amplitude (generalized coordinates) of the j-th OPO pulse measured by the homodyne detector. In the local field, H(X) is a Heaviside step function that takes 0 when X≦0 or +1 when X>0. r _j is the solution provided by CDP 104; During support vector estimation on CIM 102, all r _j are fixed. Therefore, the first term is an interaction term, and the second term corresponds to a Zeeman term.

CDP 104 obtains solutions to linear simultaneous equations from the minimization condition of H with respect to r. Without loss of generality, the element-wise expression of the simultaneous equations for the unknown values of r _i is
(7)
(8)
It can be rewritten as . To eliminate ambiguity, r _i is set to zero when H(X _i ) is zero. Here, h _i in Equation (8) is the same as the local field (Equation (6)) for support estimation on CIM 102. X _j is the solution provided by CIM 102. During signal estimation on CDP 104, all H(X _j ) are fixed. The solution to the simultaneous equation (equation (7)) is
r=(diag[A ^T A]+SA ^T AS-diag[SA ^T AS]) ^-1 SA ^T y
S=diag(H(X ₁ ), H(X ₂ ),..., H(X _N ))
It is.

Derivation of the Wigner Stochastic Differential Equation for CIM As shown in Figure 1B, the pump pulse is injected into the main ring cavity through the second harmonic generation (SHG) crystal. Periodically poled lithium niobate (PPLN) waveguides are highly efficient nonlinear media for optical parametric oscillators. Assume that the amplitude of the injection pump field into the main cavity is ε and the parametric coupling constant of the PPLN waveguide between the signal field and the pump field is κ. Then, the pumping Hamiltonian is
and the parametric interaction Hamiltonian is
It is. here,
and
is the annihilation operator for the intracavity pump and signal field. If the round-trip time of the ring cavity is properly adjusted to N times the pump pulse interval, N independent and identical OPO pulses can be generated simultaneously inside the cavity. The photon annihilation and production operator for the jth OPO signal pulse is:
and
Indicated by The intracavity pump field and signal field have loss factors γ _p and γ _s , respectively. If γ _p >> γ _s , the pump field can be eliminated by the slave principle: The following master equation of the density operator for a single j-th OPO signal pulse is obtained by adiabatic elimination of the pump mode:
where S=εκ/γ _p and B=κ ² /(2γ _p ) are the linear parametric gain factor and the two-photon absorption (or backconversion) rate, respectively.
indicates a Bosonic commutator.

The measurement feedback circuit shown in FIG. 1B is connected to the main cavity by extraction and injection couplers with reflection coefficients R _ex =j _ex Δt and R _in =j _in Δt, where j _ex and j _in are coarse-grained are the out-coupling and in-coupling constants, and Δt is the cavity round-trip time. Under the condition that B/γ _S <1 and vacuum fluctuations are incident on the open ports of the extraction and injection couplers, the measurement feedback circuit can be described using a Gaussian quantum model. The master equation consists of a linear loss term, a measured induced state reduction term, and a coherent feedback signal injection term.

The Fokker-Planck equation is the density operator in the master equation.
The following truncated Wigner stochastic differential equation (W-SDE) can be achieved by applying Ito's rule:
Here, j = j _ex + j _in , α _i is the complex Wigner amplitude, and ν _i is
<v _i (t)>=0, <v _i ^* (t)v _j (t')>=2δ _ij δ(t-t')
The c-number noise amplitude satisfies the following.

after that,
Introducing the saturation parameter A _s by the following scale transformation
and
By applying
is obtained, where c _i and s _i are the in-phase and quadrature-phase normalized amplitudes of the i-th OPO pulse. R in the upper equation of equation (18 ^* ). H. S. The second term of is the optical injection field with only in-phase components. p is the normalized pump rate. p=1 corresponds to the oscillation threshold of a single OPO without mutual coupling. If p exceeds the oscillation threshold (p>1), each of the OPO pulses is in the 0 phase state or the π phase state. The 0 phase of the OPO pulse is assigned to the Ising spin up-state, while the π phase is assigned to the down-state. The last terms in both the upper and lower equations (18 ^* ) represent the pump fluctuations coupled into the OPO system by the vacuum fluctuations injected from the external reservoir and the gain saturation. W _i,1 and W _i,2 are <W _i,k (t)>=0, <W _i,k (t)W _j,l (t')>=δ _ij δ _kl δ(t−t ') is an independent real Gaussian noise process. The saturation parameter A _s determines the non-linear increase (sudden rise) in the number of photons at the OPO threshold. Finally, a more general quantum model of MFB-CIM without the Gaussian approximation was derived for both discrete-time and continuous-time models.

Macroscopic equations for quantum-classical hybrid systems

To solve the macroscopic equations for the quantum-classical hybrid system, equation 18 ^* above is solved and simultaneous equation (7) is solved using statistical mechanics under the assumption that applying statistical mechanics is explained below. Therefore, W-SDE (18 ^* ) and simultaneous equation (7) share the same local field (equations (6) and (8)), which local field
(9)
can be rewritten by replacing as [x ₁ , . ．． ．． , x _N ] ^T is an N-dimensional source signal, [ξ ₁ , . ．． ．． , ξ _N ] ^T is the support vector. [n ₁ , . ．． ．． , n ^M ] ^T is M-dimensional observation noise that satisfies <n ^u >=0 and <n ^u n ^ν >=β ² δ _uν . β ² is the variance of observation noise.

Thus, CIM 102 and CDP 104 may be unified into a steady-state single mean eld system. Since the W-SDE for the i-th OPO only depends on the self-state and the local field h _i , a formal transfer function X from h _i to H(c _i ) can be introduced:
H(c _i )=X(h _i )
Substituting the formal transfer function X into equation (7), we also get
Therefore, the formal transfer function G from h _i to r _i is
r _i =X(h _i )h _i =G(h _i )
given by. Therefore, the local field can be defined in a self-consistent manner through the formal transfer function G as follows:
(Ten)
According to the SCSNA recipe, the local field h _i is a pure local field independent of the self-state H(c _i )r _i in the thermodynamic limit.
and an effective self-coupling term ΓH(c _i )r _i (called onsager reaction term (ORT)):
(11)
and Γ are determined in a self-consistent manner.
By using _the self-averaging property of such a mean field system, the redefined can be done. Finally, the following macroscopic equations are obtained using self-consistent local fields:
(12)
Here, R, Q, and U are macroscopic parameters called overlap, mean square magnetization, and susceptibility, respectively. <． 〉 _x,ξ indicates the average regarding x and ξ,
It is. G _c (z; xξ) and G _s (z; xξ) can be self-consistently determined from the following equations:
The saturation parameter A _s (defined above) diverges to an infinite limit of the amplitude of the infusion pump field ε→+∞. In the limit A _s →+∞, the following simplified macroscopic equation can be obtained:
(13)
here,
teeth,
is the effective output function obtained from Maxwell's rules, given by .

Accuracy of macroscopic equations and comparison with LASSO when β=0

To check the accuracy of the macroscopic equations, the above solutions are compared against the macroscopic equations with the solutions given by Algorithm 1 (Figures 9A-10, discussed in more detail below). Figures 2a and 2b compare the solution of the method for L0 regularization-based compression sensing, for different values of threshold η and compression ratio α (red and green solid lines).
Solutions and limits to macroscopic equations using (Equation (12))
The root mean square (RMSE) of the solution to the macroscopic equation in (Equation (13)) is shown. The diagram is
and
Also shown is the RMSE (circle with error bars) of the solution obtained using Algorithm 1 with
However, in experimental and realistic CIM
Note that it is of the same order as . The results in Figures 2a, 2b are for the case where there is no observation noise (ie β=0) and the source signal is from a half-Gaussian (+) or a Gaussian (±). To ensure that Algorithm 1 finds a solution corresponding to the near-zero RMSE condition obtained by the macroscopic equation, r is the true signal value in the alternating minimization process described above, i.e., as x°ξ Initialized. However, even in this situation, the c amplitude was always initialized as c=0 at the initial stage of support estimation in the alternating minimization process.

In the semi-Gaussian (+) case, two macroscopic states with non-zero RMSE (red solid line) and near-zero RMSE (green solid line) coexist as in the case of a CIM-based CDMA multi-user detector. On the other hand, for the Gaussian (±) case, a single macroscopic state with nearly zero RMSE (red solid line) was found. Comparing with the simulation results in Fig. 2b, the theoretical results obtained using macroscopic equation (13) are
The results were in good agreement with the results of Algorithm 1 using . In both the semi-Gaussian (+) and Gaussian (±) cases, the near-zero RMSE state (red solid line in Fig. 2b) of the macroscopic equation (13) is
The phase transition point given by the macroscopic equation (13) coincided with the phase transition point of Algorithm 1 (circle with error bar in Fig. 2b). Under these conditions, as η decreased to 0.01, the RMSE decreased monotonically and the phase transition point ac from the near-zero RMSE condition increased monotonically. on the other hand,
The theoretical result (left red solid line in Fig. 2a) obtained from the macroscopic equation (12) using
is in good agreement with the result of Algorithm 1 using (circle with error bar on the left in Fig. 2a), while the phase transition point given by macroscopic equation (12) is lower than that of Algorithm 1. . This is because η becomes small when it is Gaussian (±), as shown on the right side of FIG. 2a.

Furthermore, to compare the capabilities of CIM L0 regularization-based CS and LASSO, the RMSE profiles of LASSO using macroscopic equation (37) with the same threshold as CIM L0 regularization-based CS were calculated, and these profiles is superimposed on FIG. 2 (blue solid line). extreme
The RMSE of CIM L0 regularization-based CS at (red solid line in Fig. 2b) is lower than that of LASSO (blue solid line) at the same compressibility α and sparsity a, and the first-order phase transition point of CIM L0 regularization-based CS was higher than the first-order phase transition point of LASSO. On the other hand, when η=0.1 and 0.05,
The RMSE of CIM L0 regularization-based CS (red solid line and circles with error bars in Fig. 2a) was lower than that of LASSO (blue solid line), but when η = 0.01, CIM The RMSE of L0 regularization-based CS was higher than that of LASSO.

Phase diagram of CIM L0 regularization base CS and LASSO when β=0

Phase diagrams of the CIM L0 regularization-based CS for various values of threshold η were prepared when no observation noise was present (i.e., β=0). Figure 3a shows the first-order phase transition lines (red lines) from near-zero RMSE conditions for the semi-Gaussian (+) and Gaussian (±) cases. The phase transition line in Fig. 3a is the limit
This is the case.

As evidenced in Fig. 3a, the limit
In , the phase transition line from the near-zero RMSE state of the CIM L0 regularized base CS asymptotes to the black solid line a=α as η decreases, while the phase transition line from the near-zero RMSE state of the CIM L0 regularized base CS The RMSE decreases to zero (red line in Fig. 2b). The black solid line a=α in Figure 3a is a critical line that marks the boundary of whether the L0 minimization base CS can completely reconstruct the source signal without error for both the non-negative and signed cases. be. Therefore, the near-zero RMSE solution of the CIM L0 regularization-based CS asymptotes to the perfect reconstruction solution of the L0 minimization-based CS as η decreases.

To compare the properties of CIM L0 regularization-based CS with those of LASSO, Fig. 3b shows the phase diagram of LASSO: the blue line is the first-order phase transition line from the near-zero RMSE state for various η. be. As η decreases, the phase transition line from the near-zero RMSE state in LASSO for half-Gaussian (+) and Gaussian (±) asymptotes to two black dot lines, while the RMSE of the near-zero RMSE state in LASSO decreases to zero (blue line in Figure 2). The black dot lines in FIG. 3 are critical lines indicating whether the L1 minimization base CS can completely reconstruct the source signal without error for the non-negative case and the signed case, respectively. Therefore, the near-zero RMSE solution of LASSO asymptotes to the perfect reconstruction solution of L1 minimization-based CS as η decreases.

CIM L0 regularization base CS and LASSO have these asymptotic properties even for source signals from gamma (+) and bilateral gamma (±). Theoretically, this asymptotic property of the CIM L0 regularization-based CS is invariant to differences in the probability distributions of the source signals by applying a perturbative expansion to the macroscopic equation (13) in the limit η→+0. Please note that it was proven that Therefore, this theoretical result was confirmed numerically.

on the other hand,
When , the first-order phase transition line of the CIM L0 regularization base CS does not asymptote to the black solid line a=α. Around η=0.1, the phase transition line is closest to a=α.

The black dot dash line in Figure 3a is the limit
shows the lower bound of the first-order phase transition line of CIM L0 regularization base CS in . The lower limit line is above the critical line (black dot line) of the L1 minimization base CS when the compression rate α is low. The lower limit characteristic of FIG. 3a is satisfied even for source signals from gamma (+) and bilateral gamma (±). on the other hand,
When , no such lower bound exists.

Attraction sphere when β=0

To check the practicality of CIM L0 regularization-based CS, the gravity category of Algorithm 1 can be verified. To widen the gravitation sphere, the method could heuristically introduce a linear threshold decay, and when the minimization process was performed alternately, the threshold decreased linearly from η _init to η _end ( (See Algorithm 1 in Figures 9A-10). First, numerical experiments are performed to verify the size of the sphere of attraction for different values of the initial threshold η _init for a fixed η _end = 0.01 in the absence of observation noise (i.e., β = 0). It was implemented. As shown in Fig. 4a, the sphere of attraction tended to be widened by choosing an initial threshold η _init higher than η _end . As the compression ratio α decreased, this trend became more pronounced, especially for the Gaussian (±) case.

We then confirmed how well Algorithm 1 converged to the near-zero RMSE state given by macroscopic equation (13) when starting from the initial state r=0 for various η _init (Fig. 4b). As evidenced in Fig. 4b, when the sparseness a was lower than the lower limit of the first-order phase transition point (black dot dash line in Fig. 3a), Algorithm 1 with η _init =0.6 could solve the macroscopic equation ( 13) (red line), while Algorithm 1 could not converge to a solution for other values of η _init . Comparing with the RMSE profile of LASSO in Fig. 4b, Algorithm 1 exceeded the estimation accuracy of LASSO under almost all conditions where LASSO has small errors.

The characteristics shown in FIG. 4 are satisfied even when the source signals are from gamma (+) and bilateral gamma (±).

Performance of CIM L0 regularization-based CS and LASSO when

Furthermore, to check the practicality of the CIM L0 regularization-based CS, the accuracy and convergence of the CIM L0 regularization-based CS in the presence of observation noise (i.e.
) was verified. The optimal thresholds that would give the minimum RMSE of CIM L0 regularization-based CS and the minimum RMSE of LASSO (Figs. 5a and 6a) are for each method when β = 0.01, 0.05, and 0.1. was found to calculate the difference between their minimum RMSE (Figs. 5b and 6b) under the optimal threshold of . The minimum RMSE is obtained by performing a grid search on the set of solutions to macroscopic equations (13) and (37) within the range 0.002≦η≦0.5 at each point (a, α). It was done. These figures show the case of semi-Gaussian (+) and Gaussian (±) source signals. As shown in Figs. 5a and 6a, as β decreases, the phase transition line from the near-zero RMSE state in the CIM L0 regularization base CS below the optimal threshold changes to the critical line (black solid line) in the L0 minimization base CS. , the RMSE of CIM L0 regularization base CS under the optimal threshold decreases. As shown in Figures 5b and 6b, the RMSE of LASSO is higher than that of CIM L0 regularization-based CS under almost all conditions where LASSO has an error smaller than 0.2, and therefore, the RMSE of LASSO is higher than that of CIM L0 regularization-based CS exceeds the estimation accuracy of LASSO under the optimal threshold for each method.

Next, for the observation noise case, the output of Algorithm 1 converged to the solution to the macroscopic equation (13) when starting from the initial state r = 0 and η _init = 0.6.
It was determined using As shown in Figures 5c and 6c, near or at the phase transition point, Algorithm 1 converged to the solution of macroscopic equation (13).

The characteristics shown in Figures 5 and 6 were similar for source signals from gamma (+) and bilateral gamma (±).

Performance of CIM L0 regularization-based CS on realistic data

The performance of CIM L0 regularization-based CS and other methods was evaluated on realistic data. For evaluation, magnetic resonance imaging (MRI) data obtained from a high-speed MRI set was used. A Haar-wavelet transform (HWT) is applied to the data to generate a signal in which 79% of the HWT coefficients are spanned by a Haar basis function with a sparsity of 0.21. was set to zero (left panel of Figure 7a). The k-space data shown in the middle panel of Figure 7a was obtained by computing a discrete Fourier transform (DFT) from the signal in the left panel of Figure 7a, where 40% of the k-space data was undersampled at random red points in the central panel of Fig. 7a to generate an observed signal with a compression ratio of . The right panel of Fig. 7a shows an image with incoherent artifacts obtained by zero-filling Fourier reconstruction from randomly undersampled k-space data.

To achieve higher reconstruction accuracy from undersampled signals, a feasible optimization problem for CIM was formulated using L0 and L2 norms:
(14)
where x is the source signal, y is the k-space undersampling signal, F is the DFT matrix, S is the undersampling matrix, Ψ is the HWT matrix, and Δ is the second derivative matrix. , and γ and λ are regularization parameters. Under the variable transformation r=Ψx, the local field vector and interaction matrix for the CIM L0 regularized base CS are:
h=-Jr°H(X)+SFΨ ^T y,
J=ΨF ^T S ^T SFΨ ^T +γΨΔ ^T ΔΨ ^T
can be set as moreover,
LASSO performance that minimizes and
The performance of L1 minimization-based CS in minimizing .

Figure 7b was reconstructed from CIM L0 regularization-based CS (left panel of Figure 7b), LASSO (middle panel of Figure 7b), and L1 minimization-based CS performed in CVX (right panel of Figure 7b). Images (and RMSE) are shown. The CIM L0 regularization-based CS gave the most accurate reconstruction, as shown in the images circled in red in these panels.

The RMSE of the three methods as a function of threshold η was evaluated. As shown in Fig. 7c, the blue line with error bars is the RMSE of the CIM L0 regularization base CS obtained from 10 trials, the red line is the RMSE of the LASSO, and the circles are the L1 minimization RMSE of the base CS, the RMSE of the L1 minimized base CS is identical to LASSO in the limit η→0, as demonstrated in Fig. 3c. Due to the trade-off between detecting small non-zero elements and eliminating incoherent artifacts by thresholding, the optimal value for minimizing the RMSE of both CIM L0 regularization-based CS and LASSO exists. The RMSE of CIM L0 regularization-based CS was lower than that of other methods within a wide range of η.

9A-9B and 10 are a flowchart and more detailed pseudocode of a method 900 for L0 regularization-based compression sensing. In one embodiment, the CIM and CDP shown in FIG. 1B may be used to implement the method, although other devices may be used. Additionally, while the pseudocode of FIG. 10 is more detailed with respect to the exact process performed and represents a preferred implementation of the method, the method may vary from the pseudocode of FIG. 10 and be within the scope of this disclosure. There is sex. In one embodiment, the method 900 includes a computer system that includes a processor and a memory, and multiple lines of computer code/instructions stored in the memory and executed by the processor of the computer system to implement the method 900. 1B, the computer system is coupled to the CIM and CDP of the system in FIG. 1B. Alternatively, the FPGA and/or CDP of FIG. 1B can have multiple lines of computer code/instructions stored on and executed by the FPGA or CDP to perform method 900.

As shown in FIG. 9A, method 900 may use an M×N observation matrix A, an M-dimensional signal y, an N-dimensional support vector σ, and an N-dimensional signal vector r. The method may initialize (902) variables. In one embodiment shown in FIG. 10, the variables may be r and a threshold η=η _init . As shown in the pseudocode and later in the flowchart, the method may implement a loop for decreasing the threshold η, each loop of the loop performing two minimizations and a decrease of the threshold η.

During the loop shown in FIG. 9A, the method may minimize (904) a cost function (H shown in pseudocode in a preferred embodiment) with respect to the support vector. The method may use CIM 102 to perform this minimization. As shown in pseudocode, in a preferred embodiment, this process 904:
for 5 times the photon lifetime, or
For 200 times the photon lifetime, set σ=CIMsupport_estimation(r,η) while increasing the normalized pump rate from 0 to 1.5, initialize the c amplitude as c=0, and W− The SDE can be integrated numerically.

During the loop shown in FIG. 9A, the method may minimize (906) a cost function (H shown in pseudocode in a preferred embodiment) with respect to the real value (r) of the signal source. The method may use CDP 104 to perform this minimization. As shown in the pseudocode, in the preferred embodiment, this process 906 yields S=diag(σ) and r=(diag[A ^T A]+SA ^T AS−diag[SA ^T AS]) ⁻¹ SA ^T y Can be set.

As shown in FIG. 9B, the method may decrement (908) a threshold (η in a preferred embodiment). As shown in the pseudocode, in a preferred embodiment, this process 909 can decrement η:
The method may then determine (910) whether all of the iterations are finished. In one embodiment, the number of iterations of the loop (and the two minimizations) may be 50 as shown in FIG. If there are more iterations, the method loops back to process 904 to perform the next set of minimizations. When the process ends (and all of the iterations have been performed), the method returns (912) two results, which can be r and σ.

Effectiveness of CIM in Support Estimation In Algorithm 1 shown in Figures 9A, 9B and 10, the c amplitude is Always initialized as c=0. In this situation, the solution of Algorithm 1 matches very well to the near-zero RMSE condition of the macroscopic equations, as evidenced in Figure 2 and Supplementary Figure 1B, and therefore the simulated CIM reduces the support vector to It can be reconstructed up to theoretical limits. extreme
Note that the macroscopic equation in (Equation (13)) is equivalent to the macroscopic equation obtained from an Ising spin system with zero temperature. Therefore, the simulated CIM can reach the ground state to minimize H with respect to σ when r is fixed.

correctness of assumption

To derive the macroscopic equation (12), for 〈H(c _i )〉 of each OPO pulse, by replacing the state variables in the quadratic coefficient of the quantum noise power with the average value of the state variables, An approximate value was derived (see equation (19)). The macroscopic equations derived under this approximation are used in the actual equipment of CIM, as shown in Figures 2b, 5c, and 6c.
It has good accuracy in the value of . However, as shown in Fig. 2a, some solutions of the macroscopic equations are
was not consistent with the numerical solution of Algorithm 1 for smaller values of . Therefore, this approximation is possible in the steady state when the c amplitude grows and the mutual injection field is much larger than the noise.

The sphere of attraction and its dependence on the threshold

To widen the sphere of attraction in Algorithm 1, a linear threshold decay was heuristically introduced in which the threshold decreases linearly as the alternating minimization progresses. It was confirmed that the sphere of attraction becomes wider as a result of decreasing from a higher initial threshold η _init to a lower terminal threshold η _end (see Figure 4).

Following the definition of the injection field for each OPO pulse in equation (5), the threshold η acts as an external field that provides a negative bias for the OPO pulse to assume the downward state. By initially applying a large negative external field, almost all of the OPO pulses assume the π phase state, and thus almost all of {H(X _j )} _j=1,...,N zero at the initial stage of the conversion process. At the initial stage, the system can easily reach the ground state under strong negative bias. This is because the phase space consisting of a small number of upward state OPO pulses is simple. Then, through an alternating minimization process, the system tracks the gradual change in the ground state due to the incremental increase in the number of upward state OPO pulses by gradually expelling the negative external field. Finally, the system achieves the ground state at the terminal threshold η _end . This is a qualitative interpretation of the mechanism of broadening the sphere of attraction in Algorithm 1 by linearly lowering the threshold.

However, as evidenced in Fig. 4b, in the absence of observation noise, the system was unable to converge to a nearly zero RMSE solution beyond the lower bound line of the first-order phase transition point. As with the spin glass phase, there may be many pseudo-steady states at conditions above the lower limit line, and the system may therefore become trapped in one of the pseudo-steady states.

On the other hand, in the presence of observation noise, the system converged to a nearly zero RMSE solution even near the phase transition line, starting from the real initial condition r=0, as evidenced in Figs. 5c and 6c. It is suggested that the symmetry of the system allows the generation of a pseudo-steady state. Observation noise can break the symmetry about the pseudo-steady state.

conclusion

The above description has referred to specific embodiments for purposes of explanation. However, the above illustrative discussion is not intended to be exhaustive or to limit the disclosure to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. The embodiments were chosen and described to best explain the principles of the disclosure and its practical applications, so that those skilled in the art will be able to comprehend various embodiments of the disclosure and its various modifications. allowed to be best utilized to suit specific uses.

The systems and methods disclosed herein may be performed by one or more components, systems, servers, applications, other subcomponents, or distributed among such elements. When implemented as a system, such a system may include and/or require components such as software modules, general purpose CPU, RAM, etc., found in general purpose computers, among other things. In implementations where the innovation resides on a server, such a server may include and/or require components such as a CPU, RAM, etc., as found in a general purpose computer.

Additionally, the systems and methods herein may be accomplished by implementation using disparate or entirely different software, hardware, and/or firmware components beyond those described above. For example, with respect to such other components (e.g., software, processing components, etc.) and/or computer-readable media associated with or embodying the present invention, aspects of the innovation herein may be incorporated into a number of generic or may be implemented consistently with dedicated computing systems or configurations. Various exemplary computing systems, environments, and/or configurations that may be suitable for use with the innovations herein include routing/connectivity components, handheld or laptop devices, multiprocessor systems, microprocessor A personal computer, server, or server computing device, such as a base system, set-top box, consumer electronic device, network PC, other existing computer platform, or distributed computing environment that includes one or more of the above systems or devices. may include, but are not limited to, software or other components embodied in or on.

In some cases, aspects of the systems and methods may be accomplished or implemented by logic and/or logic instructions, including, for example, program modules executed in connection with such components or circuitry. Generally, program modules may include routines, programs, objects, components, data structures that perform particular tasks or execute particular instructions herein. The invention may equally be implemented in the context of a distributed software, computer, or circuit setup, where circuitry is connected by communication buses, circuitry, or links. In a distributed setting, controls/instructions can originate from both local and remote computer storage media including memory storage media.

The software, circuitry, and components herein may also include and/or utilize one or more types of computer-readable media. Computer-readable media can be any available media that resides on, is associated with, or can be accessed by such circuits and/or computing components. By way of example, and not limitation, computer-readable media can include computer storage media and communication media. Computer storage media includes volatile and nonvolatile media, removable and non-removable media, implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules, or other data. including. Computer storage media may include RAM, ROM, EEPROM, flash memory, or other memory technology, CD-ROM, digital versatile disk (DVD), or other optical storage, magnetic tape, magnetic disk storage, or Including, but not limited to, other magnetic storage devices or any other medium that can be used to store desired information and that can be accessed by a computing component. Communication media may include computer readable instructions, data structures, program modules, and/or other components. Additionally, communication media can include wired media, such as a wired network or a direct-wired connection; however, any such type of media herein may include transitory media. Not included. Combinations of any of the above are also included within the scope of computer-readable media.

In this description, the terms component, module, device, etc. can refer to any type of logical or functional software element, circuit, block, and/or process that can be implemented in a variety of ways. For example, the functionality of various circuits and/or blocks may be combined with each other into any other number of modules. Each module is stored on tangible memory (e.g., random access memory, read-only memory, CD-ROM memory, hard disk drive, etc.) that is read by a central processing unit to perform the functions of the innovation herein. It may further be implemented as a software program. Alternatively, a module may include programming instructions transmitted to a general purpose computer or to processing/graphics hardware by a communication carrier wave. Similarly, modules may be implemented as hardware logic circuitry that performs the functions encompassed by the innovations herein. Finally, the modules may be implemented using special purpose instructions (SIMD instructions), field programmable logic arrays, or any mixture thereof that provides the desired level of performance and cost.

As discussed herein, features consistent with this disclosure may be implemented by computer hardware, software, and/or firmware. For example, the systems and methods disclosed herein may be embodied in a variety of formats or combinations thereof, including, for example, a data processor, such as a computer, which also includes a database, digital electronic circuitry, firmware, and software. . Additionally, although some of the disclosed implementations describe particular hardware components, systems and methods consistent with the innovations herein may employ any combination of hardware, software, and/or firmware. It can be executed by Additionally, the features described above and other aspects and principles of the innovations herein may be implemented in a variety of environments. Such environments and associated applications may be specifically constructed to perform various routines, processes, and/or operations in accordance with the present invention, or may be selectively activated or reconfigured by code to provide the necessary functionality. may include a general-purpose computer or computing platform. The processes disclosed herein are not inherently related to any particular computer, network, architecture, environment, or other apparatus, and may be performed by any suitable combination of hardware, software, and/or firmware. obtain. For example, various general purpose machines may be used with programs written in accordance with the teachings of the present invention, or it may be more convenient to construct specialized devices or systems to implement the required methods and techniques. There is.

Aspects of the methods and systems described herein include logic, programmable logic devices (PLD), such as field programmable gate arrays (FPGAs), programmable array logic (PAL), programmable logic devices (PLDs), programmable array logic (PALs), etc. logic ) devices, electronically programmable logic and memory devices, and standard cell-based devices, as well as functions programmed into any of a variety of circuitry, including application-specific integrated circuits. can be executed. Some other possibilities for implementing the aspects include memory devices, microcontrollers with memory (such as EEPROM), embedded microprocessors, firmware, software, etc. Additionally, aspects may be embodied in microprocessors having software-based circuit emulation, discrete logic (sequential and combinatorial), custom devices, fuzzy (neural) logic, quantum devices, and hybrids of any of the above device types. can be done. The underlying device technology is based on various component types, such as complementary metal-oxide semiconductor (CMOS) metal-oxide semiconductor field-effect transistors (MOSFETs). ttransistor) technologies, bipolar technologies such as emitter-coupled logic (ECL), polymer technologies (eg, silicon conjugated polymers and metal conjugated polymer metal structures), analog-digital mixing, etc.

The various logic and/or functions disclosed herein may be implemented using any number of combinations of hardware, firmware, and/or software with respect to their behavior, register transfers, logic components, and/or other characteristics. It should also be noted that/or may be enabled as data and/or instructions embodied in a variety of machine-readable or computer-readable media. Computer-readable media on which such formatted data and/or instructions may be embodied include, but are not limited to, various forms of non-volatile storage media (e.g., optical, magnetic, or semiconductor storage media). , again, does not include a temporary medium. Unless the context clearly requires otherwise, the words "comprise," "comprising," and the like have an inclusive meaning as opposed to an exclusive or exhaustive meaning throughout the description. In other words, it is interpreted to mean "including, but not limited to." Words using the singular or plural number also include the plural or singular number respectively. Additionally, the words "herein," "hereunder," "above," "below" and words of similar meaning refer to this application as a whole. References are not made to any specific part of this application. When the word "or" is used in reference to a list of two or more items, it includes all of the following interpretations of the word: any of the items in the list; any of the items in the list; Covers all items and any combination of items in the list.

Although certain presently preferred implementations of the invention have been particularly described herein, variations and modifications of the various implementations shown and described herein may depart from the spirit and scope of the invention. It will be apparent to those skilled in the art to which this invention pertains that it may be done without. It is intended, therefore, that this invention be limited only to the extent required by applicable legal regulations.

Although the foregoing has been with respect to particular embodiments of the disclosure, modifications may be made to the embodiments without departing from the principles and spirit of the disclosure, and the scope of the disclosure is defined by the appended claims. As will be recognized by those skilled in the art.

Claims (20)

A hybrid system for L0 regularization-based compression sensing of a source signal, comprising:
a quantum machine configured to optimize a first parameter associated with the source signal to minimize a cost function;
and a classical machine configured to optimize a second parameter associated with the source signal to minimize the cost function. the source signal includes an N-dimensional source signal;
the first parameter includes a real value of the N-dimensional source signal;
2. The hybrid system of claim 1, wherein the second parameter includes a support vector associated with the N-dimensional source signal. the quantum machine and the classical machine are configured to alternatively perform corresponding optimizations of the quantum machine and the classical machine;
When the quantum machine optimizes the first parameter, the classical machine is configured to keep the second parameter constant;
2. The hybrid system of claim 1, wherein the quantum machine is configured to maintain the first parameter constant when the classical machine optimizes the second parameter. The hybrid system of claim 1, wherein the cost function includes a Hamiltonian cost function. 2. The hybrid system of claim 1, wherein the quantum machine comprises a coherent Ising machine. The hybrid system of claim 1, wherein the classical machine comprises a digital processor or a field programmable gate array. The hybrid system of claim 1, wherein the source signal includes a magnetic resonance imaging signal. A method for L0 regularization-based compression detection of a source signal, the method comprising:
optimizing by a quantum machine a first parameter associated with the source signal to minimize a cost function;
optimizing a second parameter associated with the source signal by a classical machine to minimize the cost function. the source signal includes an N-dimensional source signal;
the first parameter includes a real value of the N-dimensional source signal;
9. The method of claim 8, wherein the second parameter includes a support vector associated with the N-dimensional source signal. the quantum machine and the classical machine alternatively perform corresponding optimizations of the quantum machine and the classical machine;
When the quantum machine optimizes the first parameter, the classical machine maintains the second parameter constant;
9. The method of claim 8, wherein when the classical machine optimizes the second parameter, the quantum machine maintains the first parameter constant. 9. The method of claim 8, wherein the cost function includes a Hamiltonian cost function. 9. The method of claim 8, wherein the quantum machine comprises a coherent Ising machine. 9. The method of claim 8, wherein the classical machine comprises a digital processor or a field programmable gate array. 9. The method of claim 8, wherein the source signal comprises a magnetic resonance imaging signal. A method for implementing L0 regularization-based compression detection, the method comprising:
including injecting a plurality of pump pulses into a coherent Ising machine optical parametric oscillator having an optical parametric oscillator formed within a fiber ring cavity having an output coupler and an input coupler, the output coupler having a homodyne detection output and a second A method for communicating with a harmonic generation (SHG) crystal. extracting a portion of each pump pulse of the plurality of pump pulses from the fiber ring cavity by an output coupler on the fiber ring cavity;
16. The method of claim 15, further comprising measuring the extracted pulse using an optical homodyne detector. A feedback signal is computed by a field programmable gate array (FPGA) and provides said computed results to an intensity modulator (IM) and a phase modulator (PM), thereby transmitting said optical parametric signal through an input coupler on said fiber ring cavity. 17. The method of claim 16, further comprising generating an injection field for each of the oscillator (OPO) pulses. 18. The method of claim 17, further comprising solving linear simultaneous equations by the classical digital processor and transferring the solutions to the coherent Ising machine by a buffer. 19. The method of claim 18, further comprising using the solution from the classical digital processor to provide feedback pulses to the fiber ring input coupler. 20. The method of claim 19, further comprising estimating support vectors by the coherent Ising machine.

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